Question

Problems \(73-82\) are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area and circumference of a circle of radius \(13 \mathrm{~cm}\).

Short answer

Area: \(169\pi\; \text{cm}^2\), Circumference: \(26\pi\; \text{cm}\)

Step-by-step solution

- Recall the formulas for area and circumference of a circle

The area of a circle is given by the formula: \[ A = \pi r^2 \]The circumference of a circle is given by the formula: \[ C = 2\pi r \]where \( r \) is the radius of the circle.

- Substitute the radius into the area formula

Given the radius \( r = 13 \text{cm} \): \[ A = \pi (13)^2 \]Calculate the area: \[ A = 169\pi \text{ square centimeters} \]

- Substitute the radius into the circumference formula

Using the same radius \( r = 13 \text{cm} \): \[ C = 2\pi (13) \]Calculate the circumference: \[ C = 26\pi \text{ centimeters} \]

Key concepts

Area of a Circle

The area of a circle can be calculated using the formula \( A = \pi r^2 \). This formula tells us how much space the circle covers. Here, \pi \ (pi) is a mathematical constant approximately equal to 3.14159. The letter \ r \ stands for the radius of the circle, which is the distance from the center of the circle to any point on its edge.

To find the area, you simply square the radius and multiply it by \pi \. For example, if the radius of a circle is 13 cm, then the area can be calculated as follows:

\( A = \pi (13)^2 \)
\( A = 169 \pi \)

So, the area of the circle is 169 \pi \ square centimeters.

Knowing the area helps us understand the size of the circle and is useful in many applications like painting a circle on a wall or designing round objects.

Circumference of a Circle

The circumference of a circle is the distance around the edge of the circle. It is like the perimeter of a polygon. The formula to find the circumference is \( C = 2 \pi r \).

Here again, \pi \ is a constant (about 3.14159), and \ r \ is the radius. To find the circumference, you multiply twice the radius by \pi \. For a circle with a radius of 13 cm, the formula and solution looks like this:

\( C = 2 \pi (13) \)
\( C = 26 \pi \)

Thus, the circumference is 26 \pi \ centimeters.

Understanding the circumference is crucial for tasks such as wrapping a string around the circle or designing circular tracks.

Radius

The radius is a key element in circle geometry. It is the distance from the center of the circle to any point on its boundary. The radius plays a crucial role in defining other properties of the circle, such as the area and circumference.

For instance, in the formulas for the area (\( A = \pi r^2 \)) and circumference (\( C = 2 \pi r \)), you can see that the radius is a primary component. A longer radius means a larger circle, and both the area and circumference increase.

In our example, the radius is 13 cm. This means every point on the edge of the circle is 13 cm away from the center. The calculations we did for area and circumference were directly based on this radius length.

To summarize:

• The formula for the area of a circle is \( A = \pi r^2 \).
• The formula for the circumference of a circle is \( C = 2 \pi r \).
• Understanding the radius helps comprehend these essential circle properties more deeply.